**2. Mathematical Modeling**

The physical model displayed in Figure 1 is a 2-dimensional square enclosed box of size *L* packed with a water-based Al2O3-nanofluid. The stream is unsteady, incompressible and laminar. The velocity components (u, v) in Cartesian coordinates (x, y) are pointed to in Figure 1. The vertical walls of the enclosed domain have uniform temperature distributions. The horizontal barriers are thermally insulated. The gravity performances in the opposite of *y*-direction. The nanoliquid in the enclosed box is considered as a dilute liquid–solid mixture with a constant volume fraction of nanosized particles (Al2O3) distributed within the water. The nanoparticles and water are in thermal-equilibrium. The nanoliquid properties are presumed to be constant, except the density. The linear variation of density (with temperature) is given as ρ = ρ0[<sup>1</sup> − β(θ − <sup>θ</sup>0)], where β being the quantity of thermal expansion (Boussinesq approximation), θ is temperature and ρ0 is density at reference. The viscous dissipation is discounted here. The mathematical model for conservation of quantities is:

$$
\frac{\partial \mu}{\partial x} + \frac{\partial v}{\partial y} = 0 \tag{1}
$$

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial \mathbf{x}} + v \frac{\partial u}{\partial y} = -\frac{1}{\rho\_{nf}} \frac{\partial p}{\partial \mathbf{x}} + \frac{\mu\_{nf}}{\rho\_{nf}} \left( \frac{\partial^2 u}{\partial \mathbf{x}^2} + \frac{\partial^2 u}{\partial y^2} \right) \tag{2}$$

$$\frac{\partial v}{\partial t} + \mu \frac{\partial v}{\partial \mathbf{x}} + v \frac{\partial v}{\partial y} = -\frac{1}{\rho\_{nf}} \frac{\partial p}{\partial y} + \frac{\mu\_{nf}}{\rho\_{nf}} \left( \frac{\partial^2 v}{\partial \mathbf{x}^2} + \frac{\partial^2 v}{\partial y^2} \right) + \frac{(\rho \beta)\_{nf}}{\rho\_{nf}} \mathbf{g} (\theta - \theta\_0) \tag{3}$$

$$\frac{\partial \partial}{\partial t} + u \frac{\partial \partial}{\partial \mathbf{x}} + v \frac{\partial \partial}{\partial y} = \alpha\_{nf} \left( \frac{\partial^2 \theta}{\partial \mathbf{x}^2} + \frac{\partial^2 \theta}{\partial y^2} \right) + \frac{1}{\left( \rho c\_p \right)\_{nf}} \left( \frac{\partial q\_r}{\partial \mathbf{x}} + \frac{\partial q\_r}{\partial y} \right) \tag{4}$$

**Figure 1.** Physical model.

The subscript "nf" and "0" denote the nanofluid and reference state, respectively. The parameters *cp*, *g*, *p*,*t*, α, μ are specific heat, acceleration due to gravity, pressure, time, thermal diffusivity and the dynamic viscosity, respectively. The heat flux due to radiation along the x and y directions are set by *qrx* = −4σ<sup>∗</sup> 3*K* ∂θ<sup>4</sup> ∂*x* and *qry* = −4σ<sup>∗</sup> 3*K* ∂θ<sup>4</sup> ∂*y* , where σ\* is Stefan-Boltzmann constant and K is mean absorption coefficient. By Rosseland estimate for radiation (medium is optically thick), the thermal variances within the stream are reflected to be too small. Expanding θ4 about θ0 through Taylor series and neglecting the higher order terms obtained from Taylor series, θ4 is expressed as a function of temperature θ. That is,

$$
\theta^4 = \theta\_0^4 + 4\theta\_0^3(\theta - \theta\_0) + \dots
$$

Then, by approximating we get,

$$
\theta^4 \cong 4\theta\_0^{\cdot 3}\theta - 3\theta\_0^{\cdot 4}
$$

Therefore, the radiative heat flux reduces to

$$q\tau\_x = \frac{-16\sigma^\*\theta\_0^3}{3K'}\frac{\partial\theta}{\partial x} \text{ and } q\tau\_y = \frac{-16\sigma^\*\theta\_0^3}{3K'}\frac{\partial\theta}{\partial y} \tag{5}$$

Substituting Equation (5) into Equation (4), we ge<sup>t</sup>

$$\frac{\partial\theta}{\partial t} + u\frac{\partial\theta}{\partial \mathbf{x}} + v\frac{\partial\theta}{\partial y} = \left. a\_{nf} \right| \frac{\partial^2 \theta}{\partial \mathbf{x}^2} + \frac{\partial^2 \theta}{\partial y^2} \right) + \frac{1}{\left(\rho c\_p\right)\_{nf}} \frac{-16\sigma^\*\theta\_0^3}{3K'} \left(\frac{\partial^2 \theta}{\partial \mathbf{x}^2} + \frac{\partial^2 \theta}{\partial y^2}\right) \tag{6}$$

Initially, the velocity and temperature are zero. When *t* > 0, *u* = *v* = 0 except at top wall and *u* = +*U*0 (Case -1), *u* = −*U*<sup>0</sup> (Case -2), *v* = 0 on the top wall. For temperature, ∂θ∂*y* = 0 on the top and the bottom portions. The right and left walls are lower (θ = θ*c*) and higher (θ = θ*h*) temperature. below.

The properties of the nanoliquid in the current model are defined

Density:

$$
\rho\_{nf} = \rho\_f (1 - \phi) + \phi \rho\_p \tag{7}
$$

Thermal expansion coefficient:

$$(\rho\boldsymbol{\beta})\_{nf} = \left(\rho\boldsymbol{\beta}\right)\_f (1-\phi) + \phi(\rho\boldsymbol{\beta})\_p \tag{8}$$

Specific heat:

$$\left(\left(\rho c\_{\mathcal{P}}\right)\_{nf} = \left(\rho c\_{\mathcal{P}}\right)\_f (1 - \phi) + \phi \left(\rho c\_{\mathcal{P}}\right)\_p \tag{9}$$

The Maxwell formula is used for thermal conductivity:

$$k\_{nf} = k\_f \left[ \frac{2 + k\_{pf}^\* + 2\phi \binom{k\_{pf}^\* - 1}{p\_f}}{2 + k\_{pf}^\* - \phi \binom{k\_{pf}^\* - 1}{p\_f}} \right], \\ k\_{pf}^\* = \frac{k\_p}{k\_f} \tag{10}$$

The dynamic viscosity of nanoliquid (Ho et al. [35]) is calculated as:

$$
\mu\_{nf} = \mu\_f (1 - \phi)^{-2.5} \tag{11}
$$

where the subscript "*f* " and "*p*" denote base–fluid and nanoparticle, respectively. The physical constants of the water and nanoparticles (Al2O3) are available in Ref [35].

The leading equations are nondimensionalized by using the subsequent variables:

$$(\text{lL}, V) = \frac{(\text{u}, v)}{\text{l}l\_0}, T = \frac{\theta - \theta\_0}{\Delta \theta}, (\text{X}, Y) = \frac{(\text{x}, y)}{\text{L}}, \tau = \frac{t\text{l}l\_0}{\text{L}}, \cdots \text{and } P = \frac{p}{\rho\_{\text{nf}}l\text{l}\_0^2} \tag{12}$$

The consequent nondimensional model equations are

$$\frac{\partial \mathcal{U}}{\partial X} + \frac{\partial V}{\partial Y} = 0 \tag{13}$$

$$\frac{\partial \mathcal{U}}{\partial \tau} + \mathcal{U}\frac{\partial \mathcal{U}}{\partial \mathcal{X}} + V\frac{\partial \mathcal{U}}{\partial \mathcal{Y}} = -\frac{\partial P}{\partial \mathcal{X}} + \left(\frac{\mu\_{nf}}{\rho\_{nf}\nu\_f}\right)\left(\frac{1}{\text{Re}}\right)\frac{\partial^2 \mathcal{U}}{\partial \mathcal{X}^2} + \frac{\partial^2 \mathcal{U}}{\partial \mathcal{Y}^2}\tag{14}$$

$$\frac{\partial V}{\partial \tau} + \mathcal{U}\frac{\partial V}{\partial X} + V\frac{\partial V}{\partial Y} = -\frac{\partial P}{\partial Y} + \left(\frac{\mu\_{nf}}{\rho\_{nf}\alpha\_f}\right)\left(\frac{1}{\text{Re}}\right)\left(\frac{\partial^2 V}{\partial X^2} + \frac{\partial^2 V}{\partial Y^2}\right) + \frac{(\rho\beta)\_{nf}}{\rho\_{nf}\beta\_f}\text{Ri }T\tag{15}$$

$$\frac{\partial T}{\partial \tau} + \mathcal{U}\frac{\partial T}{\partial X} + V\frac{\partial T}{\partial Y} = \left(\frac{\alpha\_{nf}}{\alpha\_f}\frac{1}{\text{RePr}}\right) \left(1 + \frac{4k\_f}{3k\_{nf}}\mathcal{R}d\right) \left(\frac{\partial^2 T}{\partial X^2} + \frac{\partial^2 T}{\partial Y^2}\right) \tag{16}$$

The nondimensional quantities appearing above are the Grashof number *Gr* = (*g*β*f*Δθ*L*<sup>3</sup>)/(ν<sup>2</sup> *f* ), Radiation parameter *Rd* = (<sup>4</sup>σ<sup>∗</sup> θ0 <sup>3</sup>)/(*k fK* ), Richardson number *Ri* = *Gr*/Re2, Reynolds number Re = ( *<sup>U</sup>*0*<sup>L</sup>*)/(<sup>ν</sup>*f*) and the Prandtl number *Pr* = <sup>ν</sup>*f* /<sup>α</sup>*f* . The boundary settings are

$$\mathcal{U} = V = 0,\\ X = 0, 1 \& \mathcal{Y} = 0$$

$$\mathcal{U} = +1 \text{ (Case 1), } \& \mathcal{U} = -1 \text{ (Case 2), } \mathcal{V} = 0, \mathcal{Y} = 1$$

$$\frac{\partial T}{\partial Y} = 0 \text{ } \mathcal{Y} = 0 \text{ } \& \text{ 1} \tag{17}$$

$$T = 1 \text{ } X = 0 \text{ } \& \text{ } T = 0 \text{ } X = 1$$

when *U* = +1 indicates that the wall moves to the right-side and *U* = −1 implies that the wall moves to the left-side in its axis, respectively.

The drag coe fficient estimates the total frictional drag exerted on the wall. The drag coe fficient along the moving top wall is calculated as *C fx* = - ∂*U* ∂*Y Y* = 1, respectively. The averaged drag coe fficient is calculated as

$$\overline{\mathbb{C}f\_{\mathbb{X}}} = \bigcap\_{0}^{1} \mathbb{C}f\_{\mathbb{X}} \, d\mathbb{X}, \text{ respectively.}\tag{18}$$

The energy transport rate across the enclosed box is a vital parameter in thermal industrial applications. The local Nusselt number alongside the hot barrier of the enclosed box is defined as *Nu* = −*kn f k f* 1 + 4*k f* 3*kn f Rd* ∂*T*∂*X X* = 0. The averaged Nusselt number alongside the heated barrier is expressed as follows:

$$\overline{Nu} = \int\_0^1 Nu \, dY \tag{19}$$
